Difference between revisions of "ApCoCoA-1:Bertini.BSolve"
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| + | {{Version|1}} | ||
<command> | <command> | ||
<title>Bertini.BSolve</title> | <title>Bertini.BSolve</title> | ||
<short_description>Solves a zero dimensional homogeneous or non-homogeneous polynomial system of equations with default configurations.</short_description> | <short_description>Solves a zero dimensional homogeneous or non-homogeneous polynomial system of equations with default configurations.</short_description> | ||
<syntax> | <syntax> | ||
| − | Bertini.BSolve( | + | Bertini.BSolve(P:LIST, SysTyp:STRING) |
</syntax> | </syntax> | ||
<description> | <description> | ||
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||
| − | + | <par/> | |
| + | This function uses total degree homotopy to find all isolated solutions of a zero dimensional system of polynomial equations. It uses default configurations provided by Bertini. The system of polynomials may be homogeneous or nonhomogeneous. For homogeneous polynomial system the output will be the list of all real solutions and for nonhomogeneous system the output will be the list of all finite solutions. | ||
<itemize> | <itemize> | ||
<item>@param <em>P</em>: List of polynomials of the given system.</item> | <item>@param <em>P</em>: List of polynomials of the given system.</item> | ||
| − | <item>@param <em>SysTyp</em>: Type of polynomials in the list P. Homogeneous (<tt> | + | <item>@param <em>SysTyp</em>: Type of polynomials in the list <tt>P</tt>. Homogeneous (<tt>"hom"</tt>) or nonhomogeneous (<tt>"Nhom"</tt>).</item> |
<item>@return A list of lists containing the finite (or real) solutions of the system P.</item> | <item>@return A list of lists containing the finite (or real) solutions of the system P.</item> | ||
| Line 16: | Line 18: | ||
<example> | <example> | ||
| − | -- | + | -- An example of zero dimensional Non-Homogeneous Solving. |
-- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0. | -- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0. | ||
Use S ::= QQ[x,y]; | Use S ::= QQ[x,y]; | ||
P := [x^2+y^2-5, xy-2]; | P := [x^2+y^2-5, xy-2]; | ||
| − | SysTyp := | + | SysTyp := "Nhom"; |
-- Then we compute the solution with | -- Then we compute the solution with | ||
| Line 28: | Line 30: | ||
-- And we achieve a list of lists containing all finite solutions. | -- And we achieve a list of lists containing all finite solutions. | ||
---------------------------------------- | ---------------------------------------- | ||
| − | [[Vector(400000000000003/200000000000000, -3416759775755413/500000000000000000000000000000), | + | [ |
| − | Vector(9999999999999927/10000000000000000, 8966048861359829/1000000000000000000000000000000)], | + | [ |
| − | [Vector(2499999999999963/2500000000000000, 5007041073746771/100000000000000000000000000000), | + | Vector(400000000000003/200000000000000, -3416759775755413/500000000000000000000000000000), |
| − | Vector(249999999999999/125000000000000, -1089183184148021/25000000000000000000000000000)], | + | Vector(9999999999999927/10000000000000000, 8966048861359829/1000000000000000000000000000000) |
| − | [Vector(-9999999999999969/10000000000000000, 191792591213411/125000000000000000000000000000), | + | ], |
| − | Vector(-1999999999999999/1000000000000000, 2443331461729629/2500000000000000000000000000000)], | + | [ |
| − | [Vector(-250000000000001/125000000000000, 4347064 850996171/1000000000000000000000000000000), | + | Vector(2499999999999963/2500000000000000, 5007041073746771/100000000000000000000000000000), |
| − | Vector(-9999999999999943/10000000000000000, -2154842536286333/500000000000000000000000000000)]] | + | Vector(249999999999999/125000000000000, -1089183184148021/25000000000000000000000000000) |
| + | ], | ||
| + | [ | ||
| + | Vector(-9999999999999969/10000000000000000, 191792591213411/125000000000000000000000000000), | ||
| + | Vector(-1999999999999999/1000000000000000, 2443331461729629/2500000000000000000000000000000) | ||
| + | ], | ||
| + | [ | ||
| + | Vector(-250000000000001/125000000000000, 4347064 850996171/1000000000000000000000000000000), | ||
| + | Vector(-9999999999999943/10000000000000000, -2154842536286333/500000000000000000000000000000) | ||
| + | ] | ||
| + | ] | ||
| − | --For | + | --The elements of lists are vectors. Each vector represents a complex number. For example Vector(5000/1000,-4150/1000) |
| + | --represents the complex number 5000/1000-4150/1000i. | ||
| + | --For Bertini output files please refer to ApCoCoA directory/Bertini. | ||
</example> | </example> | ||
<example> | <example> | ||
| − | -- | + | -- An example of zero dimensional Homogeneous Solving |
-- We want to solve zero dimensional homogeneous system x^2-z^2=0, xy-z^2=0. | -- We want to solve zero dimensional homogeneous system x^2-z^2=0, xy-z^2=0. | ||
Use S ::= QQ[x,y,z]; | Use S ::= QQ[x,y,z]; | ||
M := [x^2-z^2, xy-z^2]; | M := [x^2-z^2, xy-z^2]; | ||
| − | SysTyp := | + | SysTyp := "hom"; |
-- Then we compute the solution with | -- Then we compute the solution with | ||
| Line 52: | Line 66: | ||
-- And we achieve a list of lists containing all real solutions. | -- And we achieve a list of lists containing all real solutions. | ||
---------------------------------------- | ---------------------------------------- | ||
| − | [[2190685167348543/5000000000000000, 2190685167348543/5000000000000000, 2190685167348543/5000000000000000], | + | [ |
| + | [2190685167348543/5000000000000000, 2190685167348543/5000000000000000, 2190685167348543/5000000000000000], | ||
[1237092982347763/5000000000000000, 1237092982347763/5000000000000000, -1237092982347763/5000000000000000], | [1237092982347763/5000000000000000, 1237092982347763/5000000000000000, -1237092982347763/5000000000000000], | ||
| − | [3235177805819999/100000000000000000000000000000, 9932123317905381/10000000000000000, | + | [3235177805819999/100000000000000000000000000000, 9932123317905381/10000000000000000,621807549382663/5000000000000000000000000000], |
| − | 621807549382663/5000000000000000000000000000], [3006769352985381/100000000000000000000000000000, | + | [3006769352985381/100000000000000000000000000000,1241515414738241/1250000000000000, 555981798431817/5000000000000000000000000000] |
| − | 1241515414738241/1250000000000000, 555981798431817/5000000000000000000000000000]] | + | ] |
| − | --For | + | --These are the real solutions of the system |
| + | --For Bertini output files please refer to ApCoCoA directory/Bertini. | ||
------------------------------------ | ------------------------------------ | ||
</example> | </example> | ||
| Line 69: | Line 85: | ||
<seealso> | <seealso> | ||
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
| − | <see>Bertini.BPCSolve</see> | + | <see>ApCoCoA-1:Bertini.BPCSolve|Bertini.BPCSolve</see> |
| − | <see>Bertini.BZCSolve</see> | + | <see>ApCoCoA-1:Bertini.BZCSolve|Bertini.BZCSolve</see> |
| − | <see>Bertini.BMSolve</see> | + | <see>ApCoCoA-1:Bertini.BMSolve|Bertini.BMSolve</see> |
| − | <see>Bertini.BUHSolve</see> | + | <see>ApCoCoA-1:Bertini.BUHSolve|Bertini.BUHSolve</see> |
</seealso> | </seealso> | ||
| Line 79: | Line 95: | ||
<key>bertini.bsolve</key> | <key>bertini.bsolve</key> | ||
<key>solve zero dimensional polynomial system</key> | <key>solve zero dimensional polynomial system</key> | ||
| − | <wiki-category>Package_bertini</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_bertini</wiki-category> |
</command> | </command> | ||
Latest revision as of 13:28, 29 October 2020
| This article is about a function from ApCoCoA-1. |
Bertini.BSolve
Solves a zero dimensional homogeneous or non-homogeneous polynomial system of equations with default configurations.
Syntax
Bertini.BSolve(P:LIST, SysTyp:STRING)
Description
Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.
This function uses total degree homotopy to find all isolated solutions of a zero dimensional system of polynomial equations. It uses default configurations provided by Bertini. The system of polynomials may be homogeneous or nonhomogeneous. For homogeneous polynomial system the output will be the list of all real solutions and for nonhomogeneous system the output will be the list of all finite solutions.
@param P: List of polynomials of the given system.
@param SysTyp: Type of polynomials in the list P. Homogeneous ("hom") or nonhomogeneous ("Nhom").
@return A list of lists containing the finite (or real) solutions of the system P.
Example
-- An example of zero dimensional Non-Homogeneous Solving. -- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0. Use S ::= QQ[x,y]; P := [x^2+y^2-5, xy-2]; SysTyp := "Nhom"; -- Then we compute the solution with Bertini.BSolve(P,SysTyp); -- And we achieve a list of lists containing all finite solutions. ---------------------------------------- [ [ Vector(400000000000003/200000000000000, -3416759775755413/500000000000000000000000000000), Vector(9999999999999927/10000000000000000, 8966048861359829/1000000000000000000000000000000) ], [ Vector(2499999999999963/2500000000000000, 5007041073746771/100000000000000000000000000000), Vector(249999999999999/125000000000000, -1089183184148021/25000000000000000000000000000) ], [ Vector(-9999999999999969/10000000000000000, 191792591213411/125000000000000000000000000000), Vector(-1999999999999999/1000000000000000, 2443331461729629/2500000000000000000000000000000) ], [ Vector(-250000000000001/125000000000000, 4347064 850996171/1000000000000000000000000000000), Vector(-9999999999999943/10000000000000000, -2154842536286333/500000000000000000000000000000) ] ] --The elements of lists are vectors. Each vector represents a complex number. For example Vector(5000/1000,-4150/1000) --represents the complex number 5000/1000-4150/1000i. --For Bertini output files please refer to ApCoCoA directory/Bertini.
Example
-- An example of zero dimensional Homogeneous Solving -- We want to solve zero dimensional homogeneous system x^2-z^2=0, xy-z^2=0. Use S ::= QQ[x,y,z]; M := [x^2-z^2, xy-z^2]; SysTyp := "hom"; -- Then we compute the solution with Bertini.BSolve(M,SysTyp); -- And we achieve a list of lists containing all real solutions. ---------------------------------------- [ [2190685167348543/5000000000000000, 2190685167348543/5000000000000000, 2190685167348543/5000000000000000], [1237092982347763/5000000000000000, 1237092982347763/5000000000000000, -1237092982347763/5000000000000000], [3235177805819999/100000000000000000000000000000, 9932123317905381/10000000000000000,621807549382663/5000000000000000000000000000], [3006769352985381/100000000000000000000000000000,1241515414738241/1250000000000000, 555981798431817/5000000000000000000000000000] ] --These are the real solutions of the system --For Bertini output files please refer to ApCoCoA directory/Bertini. ------------------------------------
See also
